PSCH 443 Intro to Regression Bivariate Regrassion

Introduction to Regression

• Regression is a statistical method used to model relationships between variables.

• Bivariate regression focuses on the effect of one independent variable on one dependent variable.

• Multiple regression extends this to study the effect of multiple independent variables on one dependent variable.

Understanding Variables

• Dependent Variable (Y): The outcome variable we care about and want to study.

• Independent Variable (X): The variable used to predict or explain changes in the dependent variable.

• Variability in Y is critical. We're interested in why some individuals may score high while others may score low on the dependent variable.

  • Examples include academic performance, mental health measures, etc.

The Goal of Regression Models

• The aim is to explain the variance of the dependent variable (Y).

• We explore questions such as:

  • Why do some people excel in standardized tests while others do not?

• Statistical modeling helps explain variability in response to changes in predictor variables.

Example of Predicting Success in College

• Outcome Variable: College GPA

• Predictor Variable: High School GPA

• We can model predicting college success using high school performance as a predictor.

• From simple regression (one predictor) to multiple regression (multiple predictors), the fundamental principles remain similar.

Visual Representation of Regression

• Scatter plots illustrate the variability of Y based on changes in X.

• A simple regression model defines a linear relationship.

  • Example equation: Y = A + B * X

    • A: Y-intercept (value of Y when X=0)

    • B: Slope (indicates how much Y changes as X increases)

Evaluating Models

• We need to define what a good model looks like and how it explains the data.

• Two approaches to regression:

  1. Best Predictor Approach: Identify which independent variables significantly explain the dependent variable.

  2. Causal Modeling Approach: A broader exploration of how a system of variables predicts an outcome, incorporating complex relationships (non-linear).

• Good models are determined by their ability to explain the most variability effectively.

Understanding Model Parameters

• Simple Linear Model: Y = A + B * X

• Parameters:

  • A (Intercept): Changes the position of the line up or down on the Y-axis.

  • B (Slope): Reflects how steep the line is, indicating the strength and direction of the relationship.

  • Y-hat (Ŷ): Represents predicted values of Y based on our model.

Interpreting Slope Changes

• The slope (B) indicates how much Y changes for each one-unit increase in X.

• Adjusting B changes the steepness of the line:

  • Positive B = increase in Y

  • Negative B = decrease in Y

  • B=0 results in a flat line, indicating no relationship.

Observed vs. Predicted Values

• Observed Values: Actual measurements collected in the study.

• Predicted Values (Ŷ): These values are derived from the regression model based on chosen parameters.

  • We will evaluate how well the predicted values match the observed values to assess the model's effectiveness.

• The process of determining A and B for the best predictive equation is crucial.